Rptqµ upt, xqFγupt, xq. (4.12) In this case, Theorem 4.2.1 allows to built a particular solution called the Floquet eigenvector, starting from the Perron eigenvector which correspond to constant parameters. Moreover, we can compare the associated Floquet eigenvalue to the Perron eigenvalue. The results about this problem are stated in Section 4.6.
Before treating the different examples, we explain in Section 4.2 the general method used to tackle these problems. It is based on the main result in Theorem 4.2.1 and the use of the eigenelements of the growth-fragmentation operator together with General Relative Entropy techniques. In Sections 4.3, 4.4, 4.5 and 4.6, we give the proofs of the results in Examples 1, 2, 3 and 4 respectively.
4.2 Technical Tools and General Method
4.2.1 Main Theorem
The proofs of the main theorems of this paper are based on the following result which requires to consider that τpxqis linear (i.e. ν1q.In this case, there exists a relation between a solution to Equation (4.1) with time-dependent parameters pV1ptq, R1ptqq and a solution to the same equation with parameters
pV2ptq, R2ptqq. More precisely, we can obtain one from the other by an appropriate dilation. The fol-lowing theorem generalizes the change of variable used in  to build self-similar solution to the pure fragmentation equation.
Theorem 4.2.1. Consider that Assumptions (4.4)-(4.7)are satisfied with ν1 andγ¡0.For u1pt, xq a solution to Equation (4.1)with parameters pV1, R1q, the function u2pt, xqdefined by
u2pt, xqWkptqu1 hptq, Wkptqx
eµ´0tpWpsqR1psqR2psqqds (4.13) with kγ1, is a solution to Equation (4.1)withpV2, R2qif W ¡0 andhsatisfy
W9 τ W
k pV2V1Wq, (4.14)
Conversely, ifh:R ÑR is one to one and ifu2is a solution with pV2, R2q, thenu1defined by (4.13) is a solution withpV1, R1q.
The proof of this result is nothing but easy calculation that we leave to the reader.
Remark 4.2.2. To check that h is one to one, we can take advantage that ODE (4.14)is a Bernoulli equation which can be integrated in
Wptq W0eτk´0tV2psqds 1 W0τ
0V1psqeτk´0sV2ps1qds1ds. (4.15) To tackle the different examples, we use Theorem 4.2.1 together with two techniques appropriate for this type of equations. First we recall the existence of particular solutions to the growth-fragmentation equation in the case of time-independent coefficients. They correspond to eigenvectors to the growth-fragmentation operator and we can give their self-similar dependency on parameters in the case of pow-erlaw coefficients (see [59, 17]). This dependency is the starting point which leads to Theorem 4.2.1 and it allows to built an invariant manifold for Equation (4.1) in the caseν 1.It also provides interesting properties on the moments of the solutions whenν 1.Then we recall results about the General Relative Entropy (GRE) introduced by [98, 99, 110] for the growth-fragmentation model. This method ensures that the particular solutions built from eigenvectors are attractive for suitable norms.
4.2.2 Eigenvectors and Self-similarity: Existence of an Invariant Manifold
When the coefficients of Equation (4.1) do not depend on time, one can build solutionspt, xqÞÑUpxqeΛt by solving the Perron eigenvalue problem
BxpτpxqUpxqqµpxqUpxq pF Uqpxq, x¥0, τp0qUp0q0, Upxq¥0, ´8
The existence of such elements Λ and U has been first studied by [95, 111] and is proved for general coefficients in . The dependency of these elements on parameters is of interest to investigate the existence of steady states for nonlinear problems (see [20, 17]). In the case of powerlaw coefficients, we can work out this dependency on frozen transport parameterV and death parameterR(see ). Under Assumptions (4.5)-(4.7), the necessary condition which appears in [41, 95] to ensure the existence of eigenelements isγ 1ν ¡0.Then we define adilation parameter
γ 1ν ¡0 (4.17)
and we have explicit self-similar dependencies
ΛpV, RqVkγΛp1,0qRµ and UpV;xqVkUp1;Vkxq. (4.18) The eigenvectorU does not depend onR,that is why we do not label it. Thereafter Λp1,0qandUp1;qare denoted by Λ andU for the sake of clarity. The result of Theorem 4.2.1 is based on the idea to use these dependencies to tackle time-dependent parameters. An intermediate result between the formula (4.13) and the dependencies (4.18) is given by the following corollary.
Corollary 4.2.3. Under the assumptions of Theorem 4.2.1, ifW is a solution to
k pV Wq (4.19)
upt, xqUpWptq;xqe´0tΛpWpsq,Rpsqqds (4.20) is a solution to Equation (4.1).
Proof. For ν 1,we can compute ΛΛp1,0qby integrating Equation (4.16) with µ0 againstx dx.
We obtain thanks to themass preservation ofF Λ
xUpxqdxτ ˆ 8
and so Λp1,0qτ.Thus using the dependency (4.18) we find that ΛpW,0qτ W and Equation (4.19) is nothing but a rewriting of Equation (4.14). We use this formulation (4.19) here to highlight the link with eigenelements, and because it allows after to obtain results in the cases whenν1.Now we apply Theorem 4.2.1 forV11, R10 andV2V, R2Rand we obtain that
u2pt, xqWkUpWkxqeΛhptq´0tRpsqdsUpWptq;xqe´0tΛpWpsq,Rpsqqds is a solution to Equation (4.1).
This corollary provides a very intuitive explicit solution in the spirit of dependencies (4.18). At each time t, the solution is an eigenvector associated to a parameter Wptq with an instantaneous fitness ΛpWptq, Rptqqassociated to the same parameterWptq.The function tÞÑWptqthus defined follows Vptq with a delay explicitely given by ODE (4.19).
A very useful consequence for the different applications is the existence of an invariant manifold for the growth-fragmentation equation with time-dependent parameters of the form (4.4). Let define the eigenmanifold
E: QUpW;q, pW, QqPpRq
which is the union of all the positive eigenlines associated to a transport parameter W. Then Corol-lary 4.2.3 ensures that, under the assumptions of Theorem 4.2.1, any solution to Equation (4.1) with an initial distributionu0PE remains inE for all time. Moreover the dynamics of such a solution reduces to the ODE (4.19) and this is the key point we use to tackle nonlinear problems.
Forν1,the technique fails and we cannot obtain explicit solution with the method of Theorem 4.2.1.
Nevertheless, we can still define W as the solution to ODE (4.19) and give properties of the functions defined as dilations of the eigenvector by (4.20). We obtain that the moments of these functions satisfy equations that are similar to the ones verified by the moments of the solution to the growth-fragmentation equation. In the special caseν 0,andγ1,we even obtain the same equation. More precisely, if we denote, forα¥0,
MαrWsptq: ˆ 8
xαUpWptq;xqe´0tΛpWpsq,Rpsqqdsdx, (4.22) and
Mαrusptq: ˆ 8
xαupt, xqdx, (4.23)
then we have the following result.
Lemma 4.2.4. On the one hand, if W is a solution to Equation (4.19), then the moments Mαsatisfy M9ααΛaαVMα ν1 p1αqΛbαMα γµRMα (4.24) with
Mα ν1rUs and bα: MαrUs
On the other hand, ifuis a solution to the fragmentation-drift equation, then the momentsMα satisfy M9αατ V Mα ν1 pcα1qβMα γµRMα (4.25) with
cα: ˆ 1
Proof. Thanks to a change of variable we can compute for allα¥0 MαrWsMαrUsWkαe´0tΛpWpsq,Rpsqqds so we have
M9α kα W9
WMα ΛpW, RqMα
αΛWkγ1pV WqMα ΛWkγMαµRMα
αΛV Wkpν1qMα p1αqΛWkγMαµRMα
αΛV aαMα ν1 p1αqΛbαMα γµRMα.
Integrating Equation (4.1) againstxαdx we obtain by integration by part and thanks to the Fubini
theorem α1 orα2.A consequence is the useful result given in the following corollary.
Corollary 4.2.5.In the case whenν 0, γ1andκsymmetric, the zero and first momentspM0,M1q
This Corollary allows in Section 4.6 to compare the Perron and Floquet eigenvalues not only forν 1 but also forν0, γ1.We do not have in this case a particular solution to the growth-fragmentation equation as in Corollary 4.2.3, but a particular solution of the reduced ODE system (4.26).
Proof. Forκsymmetric, we have already seen thatc0n02.Together with Assumption (4.3) which givesc11,we conclude thatpM0, M1qis solution to Equation (4.26).
Integrating Equation (4.16) againstdxandx dx we obtain Λβ
4.2.3 General Relative Entropy: Attractivity of the Invariant Manifold
The existence of the invariant manifold E is useful to obtain particular solutions to nonlinear growth-fragmentation equations. But what happens when the initial distribution u0 does not belong to this manifold ? The GRE method ensures thatE is attractive in a sense to be defined.
The GRE method requires to consider the adjoint growth-fragmentation equation
Btψpt, xqτpt, xq B
Bxψpt, xqµpt, xqψpt, xq pFψqpt, xq (4.27) whereF is the adjoint fragmentation operator
pFψqpt, xq: ˆ x
bpt, x, yqψpt, yqdyβpt, xqψpt, xq.
Ifuand v are two solutions to Equation (4.1) andψ is a solution to Equation (4.27), then we have for
WhenHis convex, the right hand side is nonpositive and we obtain a nonincreasing quantity called GRE.
In the case of time-independent coefficients, we can chose forv a solution of the formUpxqeΛt.Then, to apply the GRE method, we need a solution to the adjoint equation and such solutions are given by solving the adjoint Perron eigenvalue problem
Such a problem is usually solved together with the direct problem (4.16) and the first eigenvalue Λ is the same for the two ones (see [41, 95, 110]). Then the GRE ensures that any solutionuto the growth-fragmentation equation behaves asymptotically likeUpxqeΛt.More precisely it is proved in [99, 110] under general assumptions that
}̺01upt,qeΛtU}LppU1pφ dxq0, p¥1. (4.29) where̺0´
u0pyqφpyqdy withu0pxqupt0, xq.
Now consider Equation (4.1) whose coefficients are time-dependent. Under the assumptions of The-orem 4.2.1 and for p 1, the convergence result (4.29) can be interpreted as the attractivity of the invariant manifoldE inL1pφ dxqwith the distance
dpu,Eq: inf to one and we can define from a solutionupt, xqto Equation (4.1) the function
vphptq, xq:Wkptqupt, Wkptqxqe´0tpWpsqµRpsqqds (4.30) so, denoting byU and φthe eigenfunctions of Equation (4.31), we have
Forν 1, φis linear (see examples in ), so we can compute from (4.30)
upt, yqφpyqdy̺0Wkptqe´0tpWpsqµRpsqqds and
dpupt,q,Eq¤}̺1ptqupt,qWkUpW;q}}̺01vphptq,qU}Ñ0. (4.32) The exponential decay in (4.29) is proved in [111, 79] forp1 and for a constant fragmentation rate βpxqβ. It is also proved in  for powerlaw parameters in the norm corresponding top2 and this is the case we are interested in. Assume that the coefficients satisfy (4.5), (4.6) and (4.7) and assume also that the fragmentation kernel is upper and lower bounded
Dκ, κ¡0, zPr0,1s, κ¤κpzq¤κ. (4.33) Then a spectral gap result is proved in  forν 1 inL2pU1φ dxqand then the result is extended to bigger spaces thanks to a general method for spectral gaps in Hilbert spaces.
Theorem . Under Assumption (4.5) with ν 1, Assumptions (4.6) and (4.33), and Assump-tion (4.7)with γ Pp0,2s, there exista¯¡0 andr¯¥3 such that, for any aPp0,¯aqand anyr¥r,¯ there existsCa,r such that for anyu0PH:L2pθq, θpxqx xr,there holds
t¡0, }̺01upt,qeΛtU}H ¤Ca,r}̺01u0U}Heat. (4.34) This result is very useful for Applications 1 and 2 becauseL2pθqL1pxpq forr ¥2p 1. Moreover the exponential decay allows to prove exponential stability results for Equation (4.8) whenκis constant (see Section 4.3).